Views expressed here and the
recommendations here, are those of J. M. Cargal and do not reflect
the views of any organizations or journals to which he is
associated. (Other views are incorrect.) This site does
not take money from publishers, authors, or their agents. It is
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This is the most recent photograph of James M. Caral (used
with permission).

Edition 1.53, September 1, 2013. One book each on Information Theory, Matroids (in section on linear algebra) and General Physics.

Edition 1.52 April 1, 2012. Three books added on real analysis. One on advanced calculus. Two on combinatorics. One on group theory.
Edition 1.5 October 14, 2011: An essay:

Elements of Boolean Algebra (22 pages) Note that there is also a chapter on Boolean Algebra in the Lectures on algorithms, number theory, probability and other stuff link below.
Edition 1.49 January 26, 2009:
One book on General Advanced Mathematics. One book on General
Applied Mathematics. Three books added to Combinatorics ‒
two on Fibonacci numbers (the other is very strong on
Fibonacci numbers as well). One book on evolution.

Edition 1.4 (Jan 19, 2006): Due to the efforts of Bob Hofacker I have
added ISBN numbers to most books here. However, these
are here only as an aid. It is easy to switch them around or
have the wrong edition. Also added here are two books on
Abstract Algebra and one on Logic.

If you have not had the
prerequisites in the last two years, retake a prerequisite. The
belief that it will come back quickly has scuttled thousands of
careers.

Study every day – if you study
less than three days a week, you are wasting your time completely.

Break up your study: do problems,
rest and let it sink in, do problems; work in a comfortable
environment.

Never miss lecture.

Remember, even if you are able to
survive by cramming for exams, the math you learn will only go into
short term memory. Eventually, you will reach a level where you can
no longer survive by cramming, and your study habits will kill you.

Get another calculus book (bookstores are constantly closing out
university books, selling perfectly good texts for $5 or $7). A
second perspective always seems to help

Get a study aid-a book of the type: "calculus for absolute
morons"

Never miss class

Do not split the sequence. That is, do not take calc I at one school
and calc II at another. Probably your second teacher will use a
different approach from your first, when you have difficulty
changing horses midstream, your second teacher will blame it on your
first teacher having done an inferior job.

Most people come out of the calculus sequence with superficial
knowledge of the subject. However, the students who survive
with a superficial knowledge have always been the norm. Merely by
surviving, they have shown they are the good students. The really
good students will acquire a deeper knowledge of calculus with time
and continued study. Those that don't are not using calculus and it
is not clear why they needed to take it in the first place. Calculus, like basic algebra, is partly a course in technique. That is another reason to do all of your homework. There is technique and there is substance, and these things reinforce one another.

Delta-epsilon proofs in the initial sequence are generally a waste
and are abusive. They take time away from learning concepts that the
students can handle (and need). The time to learn delta-epsilon
proofs is in the first analysis course. Some students who could not
understand such proofs at all during the initial sequence actually
find them quite easy when they return to the subject.

People like to go from simple models and examples to abstraction
later. This is the normal way to learn.

There is nothing wrong to learning the syntax of the area before the
theory.

Too much motivation can be as bad as too little.

As you learn concepts, let them digest; play with them and study
them some more before moving on to the next concept.

When you get into a new area, there is something to be said for
starting with the most elementary works. For example, even if you
have a Ph.D. in physics, if you are trying to learn number theory
but have no knowledge of the subject go ahead and start with the
most elementary texts available. You are likely to find that you
will penetrate the deeper works more ably than if you had started
off with deeper works.

A basic principle is this: most serious students of mathematics
start to achieve depth in any given area the second time they study
it. If it has been three or four years since you had the calculus
sequence, go back and study your old text; you might be surprised by
how different (and easier) it seems (and how interesting). Often if
one comes back to a discipline after a six-month layoff (from that
discipline, not from math) it seems so different and much easier
than it was before. Things that went over your head the first time
now seem obvious.

A similar trick that is not for everyone and that I do not
necessarily recommend has worked for me. When studying a new area it
sometimes works to read two books simultaneously. That is: read a
chapter of one and then of the other. Pace the books so that you
read the same material at roughly the same time. The two different
viewpoints will reinforce each other in a manner that makes the
effort worthwhile.

Jan Gullberg was a Swedish surgeon.
When his son decided to major in engineering, Dr. Gullberg sat down
and wrote a book containing all the elementary mathematics he felt
every beginning engineer should know (or at least have at his
disposal). He then produced the book in camera-ready English. The
result is almost a masterpiece. It is the most readable reference
around. Every freshman and sophomore in the mathematical sciences
should have this book. It covers most calculus and everything up to
calculus, including basic algebra, and solutions of cubic and
quartic polynomials. It covers some linear algebra, quite a bit of
geometry, trigonometry, and some complex analysis and differential
equations, and more. A great book:

There are loads of books at many levels on mathematics for
engineers and/or scientists. The following book is as friendly as
any, and is well written. In many ways it is a companion to Gullberg
in that it starts primarily where Gullberg leaves off. (There is
some overlap, primarily basic calculus, but I for one don't think
that is a bad thing.) It covers much of the mathematics an engineer
might see in the last year as an undergraduate. Not only are there
the usual topics but topics one usually doesn't see in such a book,
such as group theory.

I might mention that Mathematical Methods for Physicists by
Arfken and Weber ( AP ) has a very
good reputation, but I can't vouch for it personally (since I have
never studied it). It is aimed at the senior level and above.

Most books on algebra are pretty much alike. For self study you
can almost always find decent algebra books for sale at large
bookstores (closing out inventory for various schools). Algebra at
this level is a basic tool, and it is critical to do many problems
until doing them becomes automatic. It is also critical to move on
to calculus with out much delay. For the student who has already
reached calculus I suggest Gullberg as a
reference.

With the preceding in mind I prefer books in the workbook format.

An excellent textbook series is the series by Bittinger published by
Aison-Wesley.

The modern calculus book (now the standard or traditional
model) starts with the two volume set written in the 20's by Richard
Courant. (The final version of this is Courant
and John). Most modern calculus texts (the standard
model) are remarkably alike with the shortest one in popular
use being Varburg/Parcell (Prentice-Hall: 0-13-081137-8) (post
1980 volumes tend to be more than 1000pp!). You can often find one
on sale at large bookstores (which are constantly selling off books
obtained from college bookstores).

If one standard calculus text really stands out for quality of
writing and presentation it would be:

Another book, that is standard in format and but may not be the
best for most students just beginning calculus, is the one by
Spivak. If you want to have one book to review elementary
calculus this might be it. It is an absolute favorite amongst
serious students of calculus and nerds everywhere.

Spivak, Michael. Caculus, 3rd ed.
Publish or Perish. 0-914098-89-6

Beginning students might find it as good as Simmons though.

The reformed calculus text movement is best typified by the work
of the Harvard Calculus Consortium:

However, I am not at all sold on this as a good start to
calculus. I suspect it might be useful for reviewing
calculus.

There is another unique treatment that does a great job of
motivating the material and I recommend it for students starting
out. This book is also particularly good for students who are
restudying the topic. It is an excellent resource for teachers (and
is around 600 pages):

Strang, Gilbert. Calculus. Wellesley-Cambridge.
09614-0882-0

Still another book that the beginning (serious) student might
appreciate, by one of the masters of math history is:

There are books on elementary calculus that are great when you
have already had the sequence. These are books for the serious
student of elementary calculus. The MAA series below is great
reading. Every student of the calculus should have both volumes.

The following book is simply a great book covering basic
calculus. It could work as a supplement to the text for either
the teacher or the student. It is one of the first books in a
long time to make significant use of infinitesimals without using
non-standard analysis (although Comenetz is clearly familiar with
it). I think many engineers and physicists would love
this book.

There are a lot of subtle points to his treatment. He does a nice
job of introducing a surprising number of the key ideas in the
first chapter. I think somehow that this has a great pedagogical
payoff. Although it is very similar to many other texts, I like
this particular text a great deal. Personally though I
prefer the introductory text by Strang

If choosing a text for a sophomore level course, I myself would
choose the book by Lay or the one by Strang (Wellesley-Cambridge
Press).

The following book has merit and might work well as an adjunct book
in the basic linear algebra course. It is the book for the
student just learning mathematics who wants to get into computer
graphics.

Most standard calculus texts have a section on multivariable
calculus and many sell these sections as separate texts as an
option. For example the Harvard Calculus Consortium mentioned
in Calculus sell their multivariable volume
separately.

The most informal treatment is the
second half of a series. This is a great
book for the student in third semester calculus to have on the side.

Adams, Colin, Abigail Thompson and Joel Hass. How to Ace the
Rest of Calculus: the Streetwise Guide. Freeman. 2001.
07167-4174-1

The following text is a true coffee table book with beautiful
diagrams. It uses a fair bit of linear algebra which is presented in
the text, but I suggest linear algebra as a prerequisite. Its
orientation is economics, so there is no Divergence Theorem or
Stokes Theorem.

Like in some other areas, many books on differential equations
are clones. The standard text is often little more than a cookbook
containing a large variety of tools for solving d.e.'s. Most people
use only a few of these tools. Moreover, after the course, math
majors usually forget all the techniques. Engineering students on
the other hand can remember a great deal more since they often use
these techniques. A good example of the standard text is:

The following book is the briefest around. It covers the main
topics very succinctly and is well written. Given its very modest
price and clarity I recommend it as a study aid to all students in
the basic d.e. course. Many others would appreciate it as well.

The following two volumes are exceptionally clear and well
written. Similar to the Kostelich and Armruster volume above these
emphasize geometry. These volumes rely on the geometrical view all
the way through. Note that the second volume can be read
independently of the first.

Three elementary books follow. The second and third seem to
be particularly suited as texts at the sophomore-junior level. They
emphasize linear algebra whereas Acheson is more differential
equations and physics.

There are two fantastic books that almost make a library by themselves. These are big and sumptious. The first is a solid course in undergraduate real analysis. The second is graduate level. To some extent they are available for download at their authors' web site.

This book is unusual amongst its kind for its inclusion of
applications.

There are two books for the serious student of real analysis by
Bressoud. These are books I recommend to grad students and
faculty; but one is at the undergraduate level. Very good on
history and motivation. Exceptional!!!!!

Comparable to Bressoud's books there is another historical book
on analysis that I have found readable, informative and useful (for
example I think the short chapter on Lebesgue is a good introduction
to Lebesgue theory). I like it a lot.

One of the most popular texts currently (2004) that does a nice
job for a first course is by Abbott. It does not do as much
hand holding as Bryant, which is arguably too much. It appears
to designed for a one-semester course, though you could probably
squeeze it into two semesters (with no difficulty at most
universities). Might be a nice resource for the student taking
the two-semester sequence out of another text. Minimal
pre-requisites.

Abbott, Stephen. Understanding Analysis.
Springer. 2001. 0387950605

A remarkably similar book to Abbott is the one by Pedrick.
Is even briefer, but could probably fit into two semesters at most
schools.

Pedrick, George. A First course in Analysis.
Springer. 1994. 0387941088

A more complete book at that level (more than two semesters in
my slow teaching) is

The following book is very well written it covers much of analysis
into Lebesgue measure. The chapter are short and break the material
into digestible chunks making the book a great reference, study
guide and first rate text. This may be the least appreciated book on
analysis.

Lastly any graduate student serious about analysis should also
have Korner .

The Mathematical Association of America publishes many works that
are intended as aids to teaching either calculus or analysis.
I do not know if these books are so useful to the teacher, but
they are great resources for the serious student. A recent
example is (that is particularly good):

Brabenec, Robert L. Resources for the
Study of Real Analysis. MAA.
2004. 0883857375

In general there are plenty of good books on vectors with the
two books above being outstanding. Books on differential forms and
tensors can often merely enhance the reputations of those areas for
being difficult. However, there are exceptions.

On tensors I like two books which complement each other well. The
book by Danielson is more application oriented. If you are serious
about this area get both books. Also, the Schaum outline series
volume on tensors has merit.

This book is regarded very highly by many students and researchers
for its clarity of writing and presentation. (Also, this
demonstrates how completely impartial I am, since Professor Boas
detests me.)

A tour de force at the graduate level; a book for the serious
student:

I think that a fantastic book for teaching modelling is the one that follows. It covers all sorts of modelling and is superb at the sophomore/junior level.

Shiflet, Angela B. and George W. Shiflet. Introduction to Computational Science: Mdeling and Simulation for the Sciences. Princeton University Press. 2006. 978-0691125657.

Courant and JohnA
great reference is the last edition of Courant's great classic work
on calculus. This is two volumes stretched to three with
Volume II now becoming Volume II/1 and Volume II/2.
Nonetheless they are relatively not expensive and they are great
references. Volume I is a superb work on analysis.
Volume II/1 and the first part of Volume II/2 are a full course on
multivariable calculus. Volume II/2 constitutes a great text
on applied math including differential equations, calculus of
variations, and complex analysis.

I recommend other books by Davis and Hersh as well as books by
Davis and Hersh each alone.

The late Morris Kline wrote several good books for the layman
(as well as for the professional). My personal favorite is strong on
history and art and I think deserves more attention than it has ever
had. I think it is more important now then when it was first
published (in the 1950's):

A. K. Dewdney wrote a book of 66
chapters to briefly and succinctly cover the interesting topics of
computer science. The emphasis here is theory. This is a book every
computer science major should have, and probably every math major
and certainly anyone with a serious interest in computer science.

Dewdney, A.
K. The New Turing Omnibus. Freeman. 1993. 0716782715

A nice introduction that is good at
introducing the concepts and philosophy of computer algorithms is

The serious student who wants to
specialize in combinatorics should not specialize too much. In
particular you should take courses in number theory and probability.
Abstract algebra, linear algebra, linear programming-these and other
areas can be useful.

There are two books that are extremely good one-volume introductions at the undergraduate level. Tehy are very well written. I said in printed review that book by Mazur is the best book ever published on combinatorics, or something like that. The second book compares quite favorably. They are both junior-senior level.

Here are four books at roughly the
junior-senior level. These books are all readable and are
selective in their topics. By this I mean they avoid the too
common approach of throwing in everything including the kitchen
sink.

This is a great book! Its
level is roughly senior to graduate school. (It is divided into
undergraduate and graduate halves.)

A classic text at the senior/graduate level that covers
lattices, generating functions, matroids, incidence functions and
other stuff

Aigner, Martin. Combinatorial Theory.
Springer. 1997. 3-540-61787-6

The majority of standard texts on Discrete mathematics can be
quite uninspiring. If I have to pick a single junior-senior text
that is fairly conprehensive and seems designed for the classroom
(with like most such texts enough material for at least two
semesters) I would choose:

The following two books are at an undergraduate level but of
interest to many professionals. They are both good reads and they
overlap a number of disciplines, but arguably belong most to
combinatorics. Note they do not belong in Foundations
like the book by Ebbinghaus, H.-D. Et al. The book by Bunch is
excellent for the serious freshman-sophomore. The second book is
more advanced and includes a nice treatment of Conway's own surreal
numbers.

Bunch, Bryan. The Kindom of Infinite Number: A Field Guide.
Freeman. 2000.

Most books on numerical analysis are
written to turn off the reader and to encourage him or her to go
into a different, preferably unrelated, field. Secondly, almost all
of the books in the area are written by academics or researchers at
national labs, i.e. other academics. The kind of industry I use to
work in was a little different than that. The problem is partly
textbook evolution. I've seen books long out of print that would
work nicely in the classroom. However, textbook competition requires
that newer books contain more and more material until the book can
become rather unwieldy (in several senses) for the classroom. The
truth is that the average book has far too much material for a
course. Numerical analysis touches upon so many other topics this
makes it a more demanding course than others.

A marvelous exception to the above is
the book by G. W. Stewart. It avoids the problem just mentioned
because it is based upon notes from a course. It is concise and
superbly written. (It is the one I am now teaching out of.)

This is about as elementary as I can
find. This is the problem with teaching the course. On the flip
side of course, it covers less material (e.g. fixed point
iteration is not covered). Also, it does not give pseudo-code for
algorithms. This is okay with me for the following reasons. Given
a textbook with good pseudo-code, no matter how much I lecture the
students on its points and various alternatives, they usually copy
the pseudocode as if it the word of God (rather than regarding my
word as the word of God). It is useful to make them take the
central idea of the algorithm and work out the details their
selves. This text also has an associated instructors guide and
student guides. It refers also to math packages with an emphasis
on MAPLE and a disk comes with the package, which I have ignored.

The best book on Fourier analysis
is the one by Korner. However, it is roughly at a first year
graduate level and is academic rather than say engineering
oriented. Any graduate student in analysis should have this book.

Korner, T.
W. Fourier Analysis. Cambridge. 1990. 0521389917

My favorite work on Fourier
analysis (other than Korner) is by a first rate electrical
engineer:

Number theory is one of the oldest
and most loved mathematical disciplines and as a result there have
been many great books on it. The serious student will also need to
study abstract algebra and in particular group theory.

Let me list four superb introductions.
These should be accessible to just about anyone. The book by
Davenport appears to be out of print, but not long ago it was being
published by two publishers. It might return soon. The second book
by Ore gives history without it getting in the way of learning the
subject.

Burton is not the most
elementary. He gets into arithmetic functions before he
does Euler's generalization of Fermat's Little Theorem.
However, many of the proofs are very nice. I like this one
quite bit. Like Rosen, the later editions are indeed
better.

An Introductory Text that has a
lot going for it is the one by Stillwell. It has great
material but is too fast for most beginners. Should require
a course in abstract algebra. Maybe the best second book
around on number theory.

A book I like a lot is the one by
Anderson and Bell. Although they give the proper definitions
(groups on p. 129), I recommend it to someone who already has had a
course in abstract algebra. It has applications and a lot of
information. Well laid out. Out a very good book to have.

Anderson, James A. and James M.
Bell. Number Theory with
Applications.
P-H . 1997. 0131901907

The following text makes for a
second course in number theory. It requires a first course in
abstract algebra (it often refers to proofs in Stewart's Galois
Theory which is listed in the next section (Abstract
Algebra)).

Analytic Number Theory is a tough
area and it is an area where I am not the person to ask.
However, in the early 2000's there appeared three popular books on
the Riemann Hypothesis. All three received good reviews.
The first one (Derbyshire) does the best job in explaining the
mathematics (in my opinion). Although the subject is tough
these books are essentially accessible to anyone.

Herstein was one of the best
writers on algebra. Some would consider his book as more difficult
than Fraleigh, though it doesn't go all the way through Galois
Theory (but gets most of the way there). He is particularly good (I
think) on group theory.

Hernstein has a great book on
abstract algebra at the graduate level. It is thorough, fairly
consise and beautifully written. He is very strong on motivation
and explanations. This is a four-star book (out of four stars). It
is one of the best books around on group theory. His treatment
there I think should be read by anyone interested in group theory.

Herstein,
I. N. Topics in Algebra, 2nd. ed. Wiley. 1975.
1199263311

The book by
Childs covers quite a bit of number theory as well as a whole
chapters on applications. It is certainly viable as a text, and I
definitely recommend it for the library.

The following text may be the best
two-semester graduate text around. Starting with matrix theory it
covers quite a bit of ground and is beautifully done. I like it a
great deal. Note that some people consider this book undergraduate
in level.

The following book intends to shed
light on Wiles's proof of Fermat's Last Theorem. Supposedly
it is aimed at an audience with minimal mathematics, but it should
be enlightening to students who have had a course in Abstract
Algebra who might find it fascinating.

Virtually all books on abstract
algebra and some on number theory and some on geometry get into
group theory. I have indicated which of these does an exceptional
job (in my opinion). Here we will look at books devoted to group
theory alone.

This is my favorite introductory
treatment. However, if you are comfortable with groups, but are
not acquainted with graphs of groups (Cayley diagrams) get this
book. Graphs give a great window to the subject.

The MAA published a lavish book that seems to be designed to supplant Grossman and Magnus (just above this). I prefer Grossman and Magnus for their conciseness for the elementary material. Howeever, the newer book is dazzling. It spends a long time motivating the group concept emphasizing the graphical and other visual approaches. The second part goes much deeper than Grossman and Magnus and in particular gives maybe the best treatment of the Sylow theorems that I have seen.

The next book is an introduction
that goes somewhat further than the Grossman book. It is quite
good. I think it needs a second edition. The first few sections
strike me as a little kludgy (I know, there should be a better
word-but how much am I charging you for this?) and might give a
little trouble to a true beginner.

I like this a lot. I think this
is the best on undergraduate group theory. Would be a good text
(does anyone have an undergraduate course in group theory?)

Humphreys, John F. A
Course in Group Theory. Oxford. 1996.
0198534590

This appears to be a standard
reference in much of the elementary literature.

A rather obscure book that
deserves some attention; despite the title, this book is more
groups than geometry (there are books on groups and geometry in the
geometry section). Also, it has some material on rings and the
material on geometry is non-trivial. It is very good on group
theory. Excellent at the undergraduate level for someone who has
already had exposure to groups.

If I were to recommend just one
book on geometry to an undergraduate it would probably be

Stillwell, John. The
Four Pillars of Geometry. Springer. 2005.
0-387-25530-3

An even more recent book by
Stillwell that can be classified as geometry is the following.
It recapitulates parts of several of his earlier works and is a
great pleasure to read (even if you have read the others). It
might make sense to read this first and then Four Pillars
(immediately above).

Another rather extensive book by
an authority second only to Coxeter is:

Pedoe, Dan. Geometry: A
Comprehensive Course. Dover. 1970. 0486658120

The title is correct; this book
makes for a comprehensive course, and in my view does it better
than does the book by Coxeter.

A less ambitious but readable work
is:

Roe, John. Elementary
Geometry. Oxford. 1994. 0198534566

It covers affine and projective
geometries (only a little on projective), traditional analytic
geometry a little beyond a thorough treatment of the conics. The
last two chapters cover volume and quadric respectively. This is
a very viable text for an undergraduate course.

The following two books are
intended as undergraduate texts. Both volumes are slim and do a
short course on Euclidean geometry and the development of
non-Euclidean geometry followed by affine and projective
geometries. The book by Sibley touches on a few other topics and
may be a little easier to read. I believe it was influenced heavily
by Cederberg's text. The design is very similar. She is better on
projective geometry though; I suspect he will touch that up for a
second edition. Also, when he does iterated fractal systems in 2 or
3 pages-I am not sure that that is worth the effort; do it
thoroughly or leave it.

The four books listed above are
all excellent! but there is a new book on the same topic, by a
great geometer, that I think is a masterpiece. If this topic
(traditional Euclidean geometry and the development of
non-Euclidean geometry) interests you, then you want the damn book.

The second book, in particular,
does stray from projective geometry a little.

The following books emphasize an
analytic approach. Note, this is the mathematics that lies under
computer graphics. I like the book by Henle a great deal.
Note also that the analytic approach is treated nicely in the books
by Sibley, Cederberg, and Bennett.

The following book emphasizes the
connections between affine and projective geometries with algebra.
I think that the reader should have some experience with these
geometries and with abstract algebra.

Like most books with elementary
intentions, it may require more than it claims. Yes it provides
the basic definitions of abstract algebra, but I would recommend
a course in abstract algebra before reading this book.

By set theory, I do not mean the
set theory that is the first chapter of so many texts, but rather
the specialty related to logic. See the section on
Foundations as there are books there
with a significant amount of set theory.

There is a celebrated treatment for
all readers of Gödel's Incompleteness Theorem. This book
received a Pulitzer and was a significant event. (More concisely,
the book received a lot of hype and derserved it.)

The four volumes of D. E. Knuth,
The Art of Computing, Aison-Wesley are more or less
a bible. They are comprehensive, authoritative, brilliant. They are
mathematically sophisticated and are considered by most people to be
references more than texts.

For algorithms on optimization and
linear programming and integer programming go to the appropriate
sections.

The best single book on the subject is
the one by Cormen, Leiseron, and Rivest. It covers a great deal of
ground; it is well organized; it is well written; it reviews
mathematical topics well; it has good references; the algorithms are
stated unusually clearly.

The second edition will include
recommendations on books on Digital Filters and Signal Analysis

Probability

The books listed here are all
calculus based except for the book by Bennett..

An absolutely superb book for the
layman, and of interest to the professional accomplishes what many
other books have merely attempted.

Bennett,
Deborah J. Randomness. Harvard. 1998. 0674107454

This book can instill the layman
reader with a better understanding of the nature of statistics
than the usual course in statistics for sophomores (which usually
fails miserably to do this). See also Tanur.

An interesting book, quite
philosophical, on randomness is the one by Taleb.

One of the best books written for
the undergraduate to learn probability is the book by Gordon.
Despite the restriction to discrete probability this book is a
superb general introduction for the math undergraduate and is very
well organized. Great text!!

As a rule I think that the best
books to learn probability from are those on modeling. For example,
perhaps the best writer on probability is Sheldon Ross. But I think
a better book to learn probability from than his fine A First
Course in Probability is

The following is an inexpensive
little reference. It requires only a basic knowledge of probability,
say through Bayes' Theorem. The great thing about it is that the
problems are actually interesting. I have found this to be a good
source for classroom examples.

Some books in this area are better
than others. By in large though, it is a lot of bull about ad hoc,
not particularly robust, algorithms. Claims of anything new and
profound are general pompous bullstuff. Fuzzy methods are trivial if
you have knowledge of probability and logic. In
my view the aspiring applied mathematician can not do better than to
study probability .

A book of practical statistics
as opposed to mathematical or theoretical statistics is the one by
Snedecor and Cochran. It is rigorous but does not use calculus. It
uses real life biological data for examples but is fascinating. It
is a wonderfully well written and clear book. A real masterpiece.
Anyone who actually does statistics should have this book. But
remember, though it does not require calculus it does require
mathematical maturity. My feeling is that if you want to use this
book but do not know calculus you should go back and take calculus.

Again, students almost invariably
get through the basic course on statistics without knowing what
statistics (the field) is and how statisitics are actually used.
This is a great book. See also Bennett.

Salsburg, David. The Lady
Tasting Tea. Freeman. 2001. 0805071342

This is a history of statistics
that is a very quick read. Without using a single formula it does
a much better job of telling the layman what statistics is about
than does the usual introductory text. It is also of interest to
the professional.

A classic applied book that is
readable and thorough and good to own is:

My favorite text on mathematical
statistics is definitely the following. It is a large text with
enough material for a senior level sequence in mathematical
statistics, or a more advanced graduate sequence in mathematical
statistics. It is very well done.

For the student who needs help in
the sophomore statistics course in business or the social sciences,
let me say first, that this site is far people with more advanced
problems. Still, I can heartily recommend the following:

In the area of design of
experiments and analysis of variance, the book by Hicks is a good
standard reference. The book by Box, Hunter and Hunter is wonderful
at exploring the concepts and underlying theory. The book by Saville
and Wood is worth considering by the serious student. Although its
mathematics is simple and not calculus based this is the way theory
was developed (and this is also touched upon in the book by Box,
Hunter, and Hunter.

For sampling theory there is
actually a non-technical introduction (sort of Sampling for
Dummies) by Stuart. The book by Thompson is for the
practitioner.

Stuart, Alan. Ideas of
Sampling, 3rded. Oxford. 1987. 0028530608

Thompson, Steven K. Sampling.
Wiley. 1992. 0471558710

I personally think that time series
analysis for forecasting is usually worthless. If forced to use time
series analysis for purposes of forecasting I almost always will use
double exponential smoothing possibly embellished with seasonal
attributes and built-in parameter adjusting. The bible of times
series analysis is Box and Jenkins. The book by Kendall and Ord is
fairly complete in its survey of methods. I like the book by
Bloomfield.

There are three-zillion decent, or
better, books on linear
programming. Let me mention
four. All these discuss the simplex method. I will soon make
recommendation(s) on interior point algorithm books (however they
are covered in Rardin ).

A very elementary book that does a
great job teaching the fundamentals (with pictures) is:

The following book on optimization
is at roughly the senior level. It is a book that I would recommend
to any student getting into optimization. I think it is a must-have
for any serious student of OR.

I am not to smitten with the books
in this area. For the second edition I will try to do better. Until
then, there is one excellent book in print. There is almost
certainly an excellent book to appear. The book by French is
excellent and is out of print and shouldn't be. The books by Conway
et al and Hadley et al were published in the sixties and are out of
print and despite that are first rate if you can get your hands on
them.

This is a new area for me. There are a lot of books giving
contradictory advice or useless advice. Investment theory is
inherently mathematical, but there is a mathematical offshoot known
as "technical analysis." I have dealt with it for
more than twenty years myself, and I consider it generally nonsense.
Some of it is as bad as astrology. The better (technical
analysis) stuff is basically a dead end, or perhaps I should say
deadly end. The book by Malkiel aresses it well.

One of the most readable books that seems to cover the topics
very well is:

David Luenberger and Sheldon Ross are great writers on
operations research and applied mathematics, and are brilliant.
Luenberger is at Stanford and Ross is at Berkeley. Their books
on investment are for anyone who has a good knowledge of
undergraduate applied math. These books could easily be the
best two books on the subject. I would say Ross is the more
elementary. Get both.

I do not claim that the next book is useful for investing.
Perhaps it should be elsewhere. It is purely philosophical and
could be viewed as the Zen meditation guide that accompanies Random
Walk (the preceding book). It is however an interesting book.

Taleb, Nassim Nicholas.
Fooled by Randomness: The
Hien Role of Chance in the Markets and in Life, 2nd
ed. Texere. 2004. 0812975219

This last work appears to present a contrary view to Random Walk
(Malkiel) but is not nearly as contrary as its title suggests.
A very interesting book. Perhaps I should have included it
with the first four.

Stein, Ben and Phil DeMuth. Yes, You Can Time the
Markets. Wiley. 2003. 0471430161

Two books about crashes (kind of). The book by Mandelbrot
is a good read. He has some major points. He can be
vague on mathematical details.

I really haven't gotten around to
this area yet. Secondly, I prefer to learn most physics from
specialized sources (for example to study mechanics, how about using
a book just on mechanics?). One series you are sure to hear about is
the great series by Feynman. Be aware, that it is probably more
useful to people who already have a knowledge of the subjects. Also,
it is a great reference. It deserves its reputation as a work of
genius, but in gneral I would not recommend it to someone just
beginning to learn physics.

There are many fine one volume
summaries of physics aimed at an audience with some knowledge of
mathematics. The following, my favorite du jour, requires a
good knowledge of basic calculus through vector calculus.

The following book is good
exposition and is strong on mechanics and a good introduction to
tensors.

Menzel, Donald H.
Mathematical Physics. Dover. 1961.
0-486-60056-4

The following book is quite remarkable. It is a brief summary of physics. It seems to require some undergraduate mathematics. It is perfect for the mathematical scientist who did not study physics but wants an overview. It is an amazing book.

There are several books for laymen
on the second law of thermodynamics. The first by Atkins is well
illustrated--basically it is a coffee table book. It is very good.
Atkins is one of the best science writers alive. The book by the
Goldsteins does a thorough job of discussing the history and
concepts of thermodynamics. It is also very good.

Also, another fine book (with
Schey) in the section on Vector
Calculus is the book by Marsden and Tromba. Unless my memory is
suffering the ravages of alcohol, the 4th
edition has a much more thorough treatment of Maxwell's equations of
electromagnetism than did the 2nd
edition.

A book for people interested in
electrical engineering and who want a single book to get them into
it is:

The reason that there are so many
expositions of relativity with little more than algebra is
that special relativity can be covered with little more
than algebra. It is however rather subtle and deserves a lot of
attention. (A literature professor would explain that the special
relativity is a nuanced paradigm reflecting in essence
Einstein's misogyny.) As to general relativity it can not be
understood with little more than algebra. Rather, it can be
described technically as a real mother-lover.

On the subject of general relativity
and covering special relativity as well, there is a magnum opus,
perhaps even a 44 magnum opus. This book is the book for any serious
student. I would imagine that graduate students in physics all get
it. It is 1279 pages long and it takes great pains to be
pedagogically sweet. Tensors and everything are explained ex vacua
(that is supposed to be Latin for out of nothing it probably
means death to the left-handed). I have trouble seeing this
all covered in two semesters at the graduate level. It is formidable
but it is also magnificent.

Similarly, if I have to pick one
book on special relativity it would the following. The only caveat
here is that there are many fine books on special relativity and
some of them are less technical. Nonetheless the book avoids
calculus.

Callahan, J. J. The
Geometry of Spacetime: An Introduction to Special and General
Relativity. S-V.
2000. 0387986413

Hartle, James B. Gravity:
An Introduction to Einstein's General Relativity.
AWL. 2003. 0805386629

Here are five excellent books that
get into general relativity. The last two (Harpaz and Hakim) are
very mathematical and in my judgement Harpaz is the more elementary
of the two. The book by Bergman is wonderfully concise and clear.

The disagreement between the
Dawkin's (The Selfish Gene and all that) crowd and Gould and
Eldredge (see below) is to me, a non-argument. Dawkin's view is
perfectly logical. It is a hardcore Darwinistic viewpoint. Arguments
that it is missing something seem to me to miss the point. In the
end some genes survive and spread and others do not. Explanations of
why are basically post hoc rationalizations. That is not at all to
say that these rationalizations are without merit, but they in no
way mitigate against Dawkin's view.

Whereas both of these books are
readable, the one by Coyne might be better for a general
audience.

The most interesting book I've seen
recently is fascinating because of its refutation of creationist
arguments on one hand and ts arguement on the other hand that
natural selection is compatible with a loving God. The author's
scholarship is impressive.

The books I know on population
genetics–some classics and some out of print–tend to be
tomes. The following, at 174 pages, is more concise. It is readable
by someone with a basic course in probability and the elementary
sequence in calculus.

Gillespie, John H. Population
Genetics: A Concise Guide. John Hopkins. 1998.
0801880084

See
also Foundations (where two of the books
have the word Philosophy
in their titles).

Feynman reportedly referred to
philosophy as "bullshit." I tend to agree although
philosophy of mathematics is important. There are good works
on it and there is serious bullshit. The following book is
delightful:

Casti, John L. The One True
Platonic Heaven. Joseph Henry Press (an imprint of the
National Academy of Sciences). 2003. 0309085470

Feynman himself has a great book on
the nature of science. Far too clear and readable for
professional philsophers.

Feynman, Richard. The
Character of Physical Law. MIT. 1965.

Another fine book on the nature of
science that is very readable and aresses recent controversies.

Ben-Ari,
Moti. Just a Theory: Exploring the Nature of
Science. Prometheus Books. 2005.

Science Studies is a new
discipline that began in Edinborough Scotland in the 1960's. It
claims to be interested in understanding the sociological workings
of science. However, practitioners explicitly assume that science
controversies are always resolved by politics and not by one theory
being actually better than another. They believe further that there
is no scientific method and the belief in such is naive. To them the
scientific method is a myth that is used by scientists as they
actually proceed through other means to achieve any consensus. Their
works invariably show that scientific results were the result of
politics and personalities and not based upon higher fundaments.
However, it is no great trick to prove a proposition when that
proposition happens to be your primary assumption!! The following
book is a brilliant scholarly work that touches upon science studies
and is the book that inspired Alan Sokal to perform his celebrated
hoax.

Gross, Paul R. and Norman Levitt.
Higher Superstition: The Academic Left and Its Quarrels with
Science. John Hopkins. 1994. 0801847664

See also articles on the Sokal
affair:

The Sokal Hoax: The Sham that
Shook the Academy. Bison Books. 2000.
0803279957